A354555 - OEIS

%I #19 Aug 06 2024 05:42:54

%S 1,1,2,1,2,2,1,2,4,4,1,2,4,6,8,1,2,4,8,12,16,1,2,4,8,14,24,32,1,2,4,8,

%T 16,28,48,64,1,2,4,8,16,30,56,96,128,1,2,4,8,16,32,60,112,192,256,1,2,

%U 4,8,16,32,62,120,224,384,512,1,2,4,8,16,32,64,124,240,448,768,1024

%N Rectangular array read by antidiagonals. T(m,n) is the number of degree n monic polynomials in GF_2[x] such that each irreducible factor in the prime factorization has multiplicity no greater than m, m>=1, n>=0.

%C Row m = 1 counts the squarefree monic polynomials of degree n in GF_2[x] which is the main diagonal of A356583.

%H Kent E. Morrison, <a href="https://citeseerx.ist.psu.edu/pdf/d980bc1fc0f75f630be96e7a829478d910109c67">Random polynomials over finite fields</a>, 1999.

%F For n>m, T(m,n) = 2^n - 2^(n-m).

%F G.f. for row m: (1/(1-2x))*Product_{n>=1}(1-x^(n(m+1)))^A001037(n).

%e 1, 2, 2, 4,  8, 16, 32,  64, 128, 256,  512,

%e 1, 2, 4, 6, 12, 24, 48,  96, 192, 384,  768,

%e 1, 2, 4, 8, 14, 28, 56, 112, 224, 448,  896,

%e 1, 2, 4, 8, 16, 30, 60, 120, 240, 480,  960,

%e 1, 2, 4, 8, 16, 32, 62, 124, 248, 496,  992,

%e 1, 2, 4, 8, 16, 32, 64, 126, 252, 504, 1008,

%e 1, 2, 4, 8, 16, 32, 64, 128, 254, 508, 1016,

%e 1, 2, 4, 8, 16, 32, 64, 128, 256, 510, 1020,

%e 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1022,

%e 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024

%t nn = 12; a[q_, r_] := 1/r Sum[MoebiusMu[r/d] q^d, {d, Divisors[r]}]; mfree =

%t Table[CoefficientList[Series[1/(1 -q t) Product[((1 - t^n) Sum[(t^ n)^l, {l, 0, m}])^a[q, n], {n, 1, nn}] /. q -> 2, {t, 0, nn}], t], {m, 1, nn}];

%t Table[Table[mfree[[m + 1 - i, i]], {i, 1, m}], {m, 1, nn}] // Flatten

%Y Cf. A001037, A356583.

%K nonn,tabl

%O 0,3

%A _Geoffrey Critzer_, Aug 18 2022